Show that the lines joining the origin to the points of intersection of the curve x²– xy + y² + 3x + 3y – 2 = 0 and the straight line $\mathrm{x}-\mathrm{y}-\sqrt{2}=0$ are mutually perpendicular.

The given curve is ${\mathrm{x}}^2-\mathrm{xy}+{\mathrm{y}}^2+3\mathrm{x}+3\mathrm{y}-2=0 ...\left(1\right)$
$\text{ the given line is }\mathrm{x}-\mathrm{y}-\sqrt{2}=0$

$\Rightarrow \mathrm{x}-\mathrm{y}=\sqrt{2}\Rightarrow \frac{\mathrm{x}-\mathrm{y}}{\sqrt{2}}=1 ...\left(2\right)$
Let the curve (1) intersect the line (2) at A, B.
Join $\overline{\mathrm{OA}},\overline{\mathrm{OB}}$
By homogenizing the curve (1) with line (2);
we get
$$\begin{gathered} \begin{array}{l}{\mathrm{x}}^2-\mathrm{xy}+{\mathrm{y}}^2+3\mathrm{x}\left(1\right)+3\mathrm{y}\left(1\right)-2(1{)}^2=0\\ \Rightarrow {\mathrm{x}}^2-\mathrm{xy}+{\mathrm{y}}^2+3\mathrm{x}\left(\frac{\mathrm{x}-\mathrm{y}}{\sqrt{2}}\right)+3\mathrm{y}\left(\frac{\mathrm{x}-\mathrm{y}}{\sqrt{2}}\right)-2{\left(\frac{\mathrm{x}-\mathrm{y}}{\sqrt{2}}\right)}^2=0\\ \\ \Rightarrow {\mathrm{x}}^2-\mathrm{xy}+{\mathrm{y}}^2+3\mathrm{x}\left(\frac{\mathrm{x}-\mathrm{y}}{\sqrt{2}}\right)+3\mathrm{y}\left(\frac{\mathrm{x}-\mathrm{y}}{\sqrt{2}}\right)-2{\left(\frac{\mathrm{x}-\mathrm{y}}{\sqrt{2}}\right)}^2=0\\ {\mathrm{x}}^2-\mathrm{xy}+{\mathrm{y}}^2+\left(\frac{3{\mathrm{x}}^2-3\mathrm{xy}}{\sqrt{2}}\right)+\frac{3\mathrm{xy}-3{\mathrm{y}}^2}{\sqrt{2}}-2\left(\frac{{\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{xy}}{2}\right)=0\\ \\ \Rightarrow {\mathrm{x}}^2-\mathrm{xy}+{\mathrm{y}}^2+\frac{3{\mathrm{x}}^2-3\mathrm{xy}+3\mathrm{xy}-3{\mathrm{y}}^2}{\sqrt{2}}-{\mathrm{x}}^2-{\mathrm{y}}^2+2\mathrm{xy}=0 \\ \Rightarrow 3{\mathrm{x}}^2-3{\mathrm{y}}^2+\sqrt{2}\mathrm{xy}=0 ...\left(3\right)\end{array} \end{gathered}$$
The equation (3) represents the pair of lines $\overline{\mathrm{OA}},\overline{\mathrm{OB}}$
Comparing equation (3) with $a{x}^2+2hxy+b{y}^2=0$
we get $\mathrm{a}=3,\mathrm{b}=-3,2\mathrm{h}=\sqrt{2}$
Now a + b = 3 – 3= 0
The lines joining the origin to the points of intersection of given curve (1) and the given lines (2) are mutually perpendicular.